a parameter of renormalized beta distribution

Percentage Accurate: 99.8% → 99.7%

Time: 7.4s

Alternatives: 11

Speedup: N/A×

• 41.1% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

\[\left(0 < m \land 0 < v\right) \land v < 0.25\]

Math

\[\begin{array}{l} \\ \left(\frac{m \cdot \left(1 - m\right)}{v} - 1\right) \cdot m \end{array} \]

FPCore

(FPCore (m v) :precision binary64 (* (- (/ (* m (- 1.0 m)) v) 1.0) m))

C

double code(double m, double v) {
	return (((m * (1.0 - m)) / v) - 1.0) * m;
}

Fortran

real(8) function code(m, v)
    real(8), intent (in) :: m
    real(8), intent (in) :: v
    code = (((m * (1.0d0 - m)) / v) - 1.0d0) * m
end function

Java

public static double code(double m, double v) {
	return (((m * (1.0 - m)) / v) - 1.0) * m;
}

Python

def code(m, v):
	return (((m * (1.0 - m)) / v) - 1.0) * m

Julia

function code(m, v)
	return Float64(Float64(Float64(Float64(m * Float64(1.0 - m)) / v) - 1.0) * m)
end

MATLAB

function tmp = code(m, v)
	tmp = (((m * (1.0 - m)) / v) - 1.0) * m;
end

Wolfram

code[m_, v_] := N[(N[(N[(N[(m * N[(1.0 - m), $MachinePrecision]), $MachinePrecision] / v), $MachinePrecision] - 1.0), $MachinePrecision] * m), $MachinePrecision]

Initial Program: 99.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \left(\frac{m \cdot \left(1 - m\right)}{v} - 1\right) \cdot m \end{array} \]

FPCore

(FPCore (m v) :precision binary64 (* (- (/ (* m (- 1.0 m)) v) 1.0) m))

C

double code(double m, double v) {
	return (((m * (1.0 - m)) / v) - 1.0) * m;
}

Fortran

real(8) function code(m, v)
    real(8), intent (in) :: m
    real(8), intent (in) :: v
    code = (((m * (1.0d0 - m)) / v) - 1.0d0) * m
end function

Java

public static double code(double m, double v) {
	return (((m * (1.0 - m)) / v) - 1.0) * m;
}

Python

def code(m, v):
	return (((m * (1.0 - m)) / v) - 1.0) * m

Julia

function code(m, v)
	return Float64(Float64(Float64(Float64(m * Float64(1.0 - m)) / v) - 1.0) * m)
end

MATLAB

function tmp = code(m, v)
	tmp = (((m * (1.0 - m)) / v) - 1.0) * m;
end

Wolfram

code[m_, v_] := N[(N[(N[(N[(m * N[(1.0 - m), $MachinePrecision]), $MachinePrecision] / v), $MachinePrecision] - 1.0), $MachinePrecision] * m), $MachinePrecision]

Alternative 1: 99.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq 5 \cdot 10^{-23}:\\ \;\;\;\;\mathsf{fma}\left(\frac{m}{v}, m, -m\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{m}{v} \cdot \left(m \cdot \left(1 - m\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (m v)
 :precision binary64
 (if (<= m 5e-23) (fma (/ m v) m (- m)) (* (/ m v) (* m (- 1.0 m)))))

C

double code(double m, double v) {
	double tmp;
	if (m <= 5e-23) {
		tmp = fma((m / v), m, -m);
	} else {
		tmp = (m / v) * (m * (1.0 - m));
	}
	return tmp;
}

Julia

function code(m, v)
	tmp = 0.0
	if (m <= 5e-23)
		tmp = fma(Float64(m / v), m, Float64(-m));
	else
		tmp = Float64(Float64(m / v) * Float64(m * Float64(1.0 - m)));
	end
	return tmp
end

Wolfram

code[m_, v_] := If[LessEqual[m, 5e-23], N[(N[(m / v), $MachinePrecision] * m + (-m)), $MachinePrecision], N[(N[(m / v), $MachinePrecision] * N[(m * N[(1.0 - m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if m < 5.0000000000000002e-23

   1. Initial program 99.8%

      \[\left(\frac{m \cdot \left(1 - m\right)}{v} - 1\right) \cdot m \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left(\frac{m \cdot \left(1 - m\right)}{v} - 1\right) \cdot m} \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{m \cdot \left(\frac{m \cdot \left(1 - m\right)}{v} - 1\right)} \]

      3. lift--.f64 N/A

         \[\leadsto m \cdot \color{blue}{\left(\frac{m \cdot \left(1 - m\right)}{v} - 1\right)} \]

      4. sub-neg N/A

         \[\leadsto m \cdot \color{blue}{\left(\frac{m \cdot \left(1 - m\right)}{v} + \left(\mathsf{neg}\left(1\right)\right)\right)} \]

      5. distribute-rgt-in N/A

         \[\leadsto \color{blue}{\frac{m \cdot \left(1 - m\right)}{v} \cdot m + \left(\mathsf{neg}\left(1\right)\right) \cdot m} \]

      6. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{m \cdot \left(1 - m\right)}{v}} \cdot m + \left(\mathsf{neg}\left(1\right)\right) \cdot m \]

      7. clear-num N/A

         \[\leadsto \color{blue}{\frac{1}{\frac{v}{m \cdot \left(1 - m\right)}}} \cdot m + \left(\mathsf{neg}\left(1\right)\right) \cdot m \]

      8. associate-/r/ N/A

         \[\leadsto \color{blue}{\left(\frac{1}{v} \cdot \left(m \cdot \left(1 - m\right)\right)\right)} \cdot m + \left(\mathsf{neg}\left(1\right)\right) \cdot m \]

      9. associate-*l* N/A

         \[\leadsto \color{blue}{\frac{1}{v} \cdot \left(\left(m \cdot \left(1 - m\right)\right) \cdot m\right)} + \left(\mathsf{neg}\left(1\right)\right) \cdot m \]

      10. metadata-eval N/A

          \[\leadsto \frac{1}{v} \cdot \left(\left(m \cdot \left(1 - m\right)\right) \cdot m\right) + \color{blue}{-1} \cdot m \]

      11. neg-mul-1 N/A

          \[\leadsto \frac{1}{v} \cdot \left(\left(m \cdot \left(1 - m\right)\right) \cdot m\right) + \color{blue}{\left(\mathsf{neg}\left(m\right)\right)} \]

      12. lower-fma.f64 N/A

          \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{1}{v}, \left(m \cdot \left(1 - m\right)\right) \cdot m, \mathsf{neg}\left(m\right)\right)} \]

      13. lower-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{1}{v}}, \left(m \cdot \left(1 - m\right)\right) \cdot m, \mathsf{neg}\left(m\right)\right) \]

      14. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{1}{v}, \color{blue}{\left(m \cdot \left(1 - m\right)\right) \cdot m}, \mathsf{neg}\left(m\right)\right) \]

      15. lift-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{1}{v}, \color{blue}{\left(m \cdot \left(1 - m\right)\right)} \cdot m, \mathsf{neg}\left(m\right)\right) \]

      16. lift--.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{1}{v}, \left(m \cdot \color{blue}{\left(1 - m\right)}\right) \cdot m, \mathsf{neg}\left(m\right)\right) \]

      17. sub-neg N/A

          \[\leadsto \mathsf{fma}\left(\frac{1}{v}, \left(m \cdot \color{blue}{\left(1 + \left(\mathsf{neg}\left(m\right)\right)\right)}\right) \cdot m, \mathsf{neg}\left(m\right)\right) \]

      18. +-commutative N/A

          \[\leadsto \mathsf{fma}\left(\frac{1}{v}, \left(m \cdot \color{blue}{\left(\left(\mathsf{neg}\left(m\right)\right) + 1\right)}\right) \cdot m, \mathsf{neg}\left(m\right)\right) \]

      19. distribute-lft-in N/A

          \[\leadsto \mathsf{fma}\left(\frac{1}{v}, \color{blue}{\left(m \cdot \left(\mathsf{neg}\left(m\right)\right) + m \cdot 1\right)} \cdot m, \mathsf{neg}\left(m\right)\right) \]

      20. *-rgt-identity N/A

          \[\leadsto \mathsf{fma}\left(\frac{1}{v}, \left(m \cdot \left(\mathsf{neg}\left(m\right)\right) + \color{blue}{m}\right) \cdot m, \mathsf{neg}\left(m\right)\right) \]

      21. lower-fma.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{1}{v}, \color{blue}{\mathsf{fma}\left(m, \mathsf{neg}\left(m\right), m\right)} \cdot m, \mathsf{neg}\left(m\right)\right) \]

      22. lower-neg.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{1}{v}, \mathsf{fma}\left(m, \color{blue}{\mathsf{neg}\left(m\right)}, m\right) \cdot m, \mathsf{neg}\left(m\right)\right) \]

      23. lower-neg.f64 90.3

          \[\leadsto \mathsf{fma}\left(\frac{1}{v}, \mathsf{fma}\left(m, -m, m\right) \cdot m, \color{blue}{-m}\right) \]

   4. Applied rewrites 90.3%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{1}{v}, \mathsf{fma}\left(m, -m, m\right) \cdot m, -m\right)} \]

   5. Taylor expanded in m around 0

      \[\leadsto \color{blue}{m \cdot \left(\frac{m}{v} - 1\right)} \]
   6. Step-by-step derivation

      1. distribute-lft-out-- N/A

         \[\leadsto \color{blue}{m \cdot \frac{m}{v} - m \cdot 1} \]

      2. associate-/l* N/A

         \[\leadsto \color{blue}{\frac{m \cdot m}{v}} - m \cdot 1 \]

      3. unpow2 N/A

         \[\leadsto \frac{\color{blue}{{m}^{2}}}{v} - m \cdot 1 \]

      4. *-rgt-identity N/A

         \[\leadsto \frac{{m}^{2}}{v} - \color{blue}{m} \]

      5. lower--.f64 N/A

         \[\leadsto \color{blue}{\frac{{m}^{2}}{v} - m} \]

      6. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{{m}^{2}}{v}} - m \]

      7. unpow2 N/A

         \[\leadsto \frac{\color{blue}{m \cdot m}}{v} - m \]

      8. lower-*.f64 90.4

         \[\leadsto \frac{\color{blue}{m \cdot m}}{v} - m \]

   7. Applied rewrites 90.4%

      \[\leadsto \color{blue}{\frac{m \cdot m}{v} - m} \]
   8. Step-by-step derivation

   9. Applied rewrites 99.8%

      \[\leadsto \mathsf{fma}\left(\frac{m}{v}, \color{blue}{m}, -m\right) \]

   if 5.0000000000000002e-23 < m

   1. Initial program 99.8%

      \[\left(\frac{m \cdot \left(1 - m\right)}{v} - 1\right) \cdot m \]
   2. Add Preprocessing

   3. Taylor expanded in m around 0

      \[\leadsto \color{blue}{m \cdot \left(m \cdot \left(-1 \cdot \frac{m}{v} + \frac{1}{v}\right) - 1\right)} \]
   4. Step-by-step derivation

      1. distribute-lft-out-- N/A

         \[\leadsto \color{blue}{m \cdot \left(m \cdot \left(-1 \cdot \frac{m}{v} + \frac{1}{v}\right)\right) - m \cdot 1} \]

      2. +-commutative N/A

         \[\leadsto m \cdot \left(m \cdot \color{blue}{\left(\frac{1}{v} + -1 \cdot \frac{m}{v}\right)}\right) - m \cdot 1 \]

      3. mul-1-neg N/A

         \[\leadsto m \cdot \left(m \cdot \left(\frac{1}{v} + \color{blue}{\left(\mathsf{neg}\left(\frac{m}{v}\right)\right)}\right)\right) - m \cdot 1 \]

      4. unsub-neg N/A

         \[\leadsto m \cdot \left(m \cdot \color{blue}{\left(\frac{1}{v} - \frac{m}{v}\right)}\right) - m \cdot 1 \]

      5. div-sub N/A

         \[\leadsto m \cdot \left(m \cdot \color{blue}{\frac{1 - m}{v}}\right) - m \cdot 1 \]

      6. associate-/l* N/A

         \[\leadsto m \cdot \color{blue}{\frac{m \cdot \left(1 - m\right)}{v}} - m \cdot 1 \]

      7. *-commutative N/A

         \[\leadsto m \cdot \frac{\color{blue}{\left(1 - m\right) \cdot m}}{v} - m \cdot 1 \]

      8. associate-/l* N/A

         \[\leadsto m \cdot \color{blue}{\left(\left(1 - m\right) \cdot \frac{m}{v}\right)} - m \cdot 1 \]

      9. associate-*r* N/A

         \[\leadsto \color{blue}{\left(m \cdot \left(1 - m\right)\right) \cdot \frac{m}{v}} - m \cdot 1 \]

      10. *-inverses N/A

          \[\leadsto \left(m \cdot \left(1 - m\right)\right) \cdot \frac{m}{v} - m \cdot \color{blue}{\frac{v}{v}} \]

      11. associate-/l* N/A

          \[\leadsto \left(m \cdot \left(1 - m\right)\right) \cdot \frac{m}{v} - \color{blue}{\frac{m \cdot v}{v}} \]

      12. *-commutative N/A

          \[\leadsto \left(m \cdot \left(1 - m\right)\right) \cdot \frac{m}{v} - \frac{\color{blue}{v \cdot m}}{v} \]

      13. associate-/l* N/A

          \[\leadsto \left(m \cdot \left(1 - m\right)\right) \cdot \frac{m}{v} - \color{blue}{v \cdot \frac{m}{v}} \]

      14. distribute-rgt-out-- N/A

          \[\leadsto \color{blue}{\frac{m}{v} \cdot \left(m \cdot \left(1 - m\right) - v\right)} \]

      15. unsub-neg N/A

          \[\leadsto \frac{m}{v} \cdot \color{blue}{\left(m \cdot \left(1 - m\right) + \left(\mathsf{neg}\left(v\right)\right)\right)} \]

      16. mul-1-neg N/A

          \[\leadsto \frac{m}{v} \cdot \left(m \cdot \left(1 - m\right) + \color{blue}{-1 \cdot v}\right) \]

      17. +-commutative N/A

          \[\leadsto \frac{m}{v} \cdot \color{blue}{\left(-1 \cdot v + m \cdot \left(1 - m\right)\right)} \]

      18. lower-*.f64 N/A

          \[\leadsto \color{blue}{\frac{m}{v} \cdot \left(-1 \cdot v + m \cdot \left(1 - m\right)\right)} \]

      19. lower-/.f64 N/A

          \[\leadsto \color{blue}{\frac{m}{v}} \cdot \left(-1 \cdot v + m \cdot \left(1 - m\right)\right) \]

      20. +-commutative N/A

          \[\leadsto \frac{m}{v} \cdot \color{blue}{\left(m \cdot \left(1 - m\right) + -1 \cdot v\right)} \]

   5. Applied rewrites 99.9%

      \[\leadsto \color{blue}{\frac{m}{v} \cdot \left(m - \mathsf{fma}\left(m, m, v\right)\right)} \]

   6. Taylor expanded in m around inf

      \[\leadsto \frac{m}{v} \cdot \left({m}^{2} \cdot \color{blue}{\left(\frac{1}{m} - 1\right)}\right) \]
   7. Step-by-step derivation

   8. Applied rewrites 99.9%

      \[\leadsto \frac{m}{v} \cdot \left(m - \color{blue}{m \cdot m}\right) \]
   9. Step-by-step derivation

   10. Applied rewrites 99.9%

       \[\leadsto \frac{m}{v} \cdot \left(\left(1 - m\right) \cdot m\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq 5 \cdot 10^{-23}:\\ \;\;\;\;\mathsf{fma}\left(\frac{m}{v}, m, -m\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{m}{v} \cdot \left(m \cdot \left(1 - m\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 97.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \cdot \left(\frac{m \cdot \left(1 - m\right)}{v} + -1\right) \leq -2 \cdot 10^{+37}:\\ \;\;\;\;\frac{m \cdot \left(m \cdot m\right)}{-v}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{m}{v}, m, -m\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (m v)
 :precision binary64
 (if (<= (* m (+ (/ (* m (- 1.0 m)) v) -1.0)) -2e+37)
   (/ (* m (* m m)) (- v))
   (fma (/ m v) m (- m))))

C

double code(double m, double v) {
	double tmp;
	if ((m * (((m * (1.0 - m)) / v) + -1.0)) <= -2e+37) {
		tmp = (m * (m * m)) / -v;
	} else {
		tmp = fma((m / v), m, -m);
	}
	return tmp;
}

Julia

function code(m, v)
	tmp = 0.0
	if (Float64(m * Float64(Float64(Float64(m * Float64(1.0 - m)) / v) + -1.0)) <= -2e+37)
		tmp = Float64(Float64(m * Float64(m * m)) / Float64(-v));
	else
		tmp = fma(Float64(m / v), m, Float64(-m));
	end
	return tmp
end

Wolfram

code[m_, v_] := If[LessEqual[N[(m * N[(N[(N[(m * N[(1.0 - m), $MachinePrecision]), $MachinePrecision] / v), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision], -2e+37], N[(N[(m * N[(m * m), $MachinePrecision]), $MachinePrecision] / (-v)), $MachinePrecision], N[(N[(m / v), $MachinePrecision] * m + (-m)), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (-.f64 (/.f64 (*.f64 m (-.f64 #s(literal 1 binary64) m)) v) #s(literal 1 binary64)) m) < -1.99999999999999991e37

   1. Initial program 99.9%

      \[\left(\frac{m \cdot \left(1 - m\right)}{v} - 1\right) \cdot m \]
   2. Add Preprocessing

   3. Taylor expanded in m around inf

      \[\leadsto \color{blue}{-1 \cdot \frac{{m}^{3}}{v}} \]
   4. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto \color{blue}{\frac{-1 \cdot {m}^{3}}{v}} \]

      2. associate-*l/ N/A

         \[\leadsto \color{blue}{\frac{-1}{v} \cdot {m}^{3}} \]

      3. metadata-eval N/A

         \[\leadsto \frac{\color{blue}{\mathsf{neg}\left(1\right)}}{v} \cdot {m}^{3} \]

      4. distribute-neg-frac N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\frac{1}{v}\right)\right)} \cdot {m}^{3} \]

      5. distribute-lft-neg-in N/A

         \[\leadsto \color{blue}{\mathsf{neg}\left(\frac{1}{v} \cdot {m}^{3}\right)} \]

      6. *-commutative N/A

         \[\leadsto \mathsf{neg}\left(\color{blue}{{m}^{3} \cdot \frac{1}{v}}\right) \]

      7. lower-neg.f64 N/A

         \[\leadsto \color{blue}{\mathsf{neg}\left({m}^{3} \cdot \frac{1}{v}\right)} \]

      8. associate-*r/ N/A

         \[\leadsto \mathsf{neg}\left(\color{blue}{\frac{{m}^{3} \cdot 1}{v}}\right) \]

      9. *-rgt-identity N/A

         \[\leadsto \mathsf{neg}\left(\frac{\color{blue}{{m}^{3}}}{v}\right) \]

      10. lower-/.f64 N/A

          \[\leadsto \mathsf{neg}\left(\color{blue}{\frac{{m}^{3}}{v}}\right) \]

      11. cube-mult N/A

          \[\leadsto \mathsf{neg}\left(\frac{\color{blue}{m \cdot \left(m \cdot m\right)}}{v}\right) \]

      12. unpow2 N/A

          \[\leadsto \mathsf{neg}\left(\frac{m \cdot \color{blue}{{m}^{2}}}{v}\right) \]

      13. lower-*.f64 N/A

          \[\leadsto \mathsf{neg}\left(\frac{\color{blue}{m \cdot {m}^{2}}}{v}\right) \]

      14. unpow2 N/A

          \[\leadsto \mathsf{neg}\left(\frac{m \cdot \color{blue}{\left(m \cdot m\right)}}{v}\right) \]

      15. lower-*.f64 98.2

          \[\leadsto -\frac{m \cdot \color{blue}{\left(m \cdot m\right)}}{v} \]

   5. Applied rewrites 98.2%

      \[\leadsto \color{blue}{-\frac{m \cdot \left(m \cdot m\right)}{v}} \]

   if -1.99999999999999991e37 < (*.f64 (-.f64 (/.f64 (*.f64 m (-.f64 #s(literal 1 binary64) m)) v) #s(literal 1 binary64)) m)

   1. Initial program 99.8%

      \[\left(\frac{m \cdot \left(1 - m\right)}{v} - 1\right) \cdot m \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left(\frac{m \cdot \left(1 - m\right)}{v} - 1\right) \cdot m} \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{m \cdot \left(\frac{m \cdot \left(1 - m\right)}{v} - 1\right)} \]

      3. lift--.f64 N/A

         \[\leadsto m \cdot \color{blue}{\left(\frac{m \cdot \left(1 - m\right)}{v} - 1\right)} \]

      4. sub-neg N/A

         \[\leadsto m \cdot \color{blue}{\left(\frac{m \cdot \left(1 - m\right)}{v} + \left(\mathsf{neg}\left(1\right)\right)\right)} \]

      5. distribute-rgt-in N/A

         \[\leadsto \color{blue}{\frac{m \cdot \left(1 - m\right)}{v} \cdot m + \left(\mathsf{neg}\left(1\right)\right) \cdot m} \]

      6. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{m \cdot \left(1 - m\right)}{v}} \cdot m + \left(\mathsf{neg}\left(1\right)\right) \cdot m \]

      7. clear-num N/A

         \[\leadsto \color{blue}{\frac{1}{\frac{v}{m \cdot \left(1 - m\right)}}} \cdot m + \left(\mathsf{neg}\left(1\right)\right) \cdot m \]

      8. associate-/r/ N/A

         \[\leadsto \color{blue}{\left(\frac{1}{v} \cdot \left(m \cdot \left(1 - m\right)\right)\right)} \cdot m + \left(\mathsf{neg}\left(1\right)\right) \cdot m \]

      9. associate-*l* N/A

         \[\leadsto \color{blue}{\frac{1}{v} \cdot \left(\left(m \cdot \left(1 - m\right)\right) \cdot m\right)} + \left(\mathsf{neg}\left(1\right)\right) \cdot m \]

      10. metadata-eval N/A

          \[\leadsto \frac{1}{v} \cdot \left(\left(m \cdot \left(1 - m\right)\right) \cdot m\right) + \color{blue}{-1} \cdot m \]

      11. neg-mul-1 N/A

          \[\leadsto \frac{1}{v} \cdot \left(\left(m \cdot \left(1 - m\right)\right) \cdot m\right) + \color{blue}{\left(\mathsf{neg}\left(m\right)\right)} \]

      12. lower-fma.f64 N/A

          \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{1}{v}, \left(m \cdot \left(1 - m\right)\right) \cdot m, \mathsf{neg}\left(m\right)\right)} \]

      13. lower-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{1}{v}}, \left(m \cdot \left(1 - m\right)\right) \cdot m, \mathsf{neg}\left(m\right)\right) \]

      14. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{1}{v}, \color{blue}{\left(m \cdot \left(1 - m\right)\right) \cdot m}, \mathsf{neg}\left(m\right)\right) \]

      15. lift-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{1}{v}, \color{blue}{\left(m \cdot \left(1 - m\right)\right)} \cdot m, \mathsf{neg}\left(m\right)\right) \]

      16. lift--.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{1}{v}, \left(m \cdot \color{blue}{\left(1 - m\right)}\right) \cdot m, \mathsf{neg}\left(m\right)\right) \]

      17. sub-neg N/A

          \[\leadsto \mathsf{fma}\left(\frac{1}{v}, \left(m \cdot \color{blue}{\left(1 + \left(\mathsf{neg}\left(m\right)\right)\right)}\right) \cdot m, \mathsf{neg}\left(m\right)\right) \]

      18. +-commutative N/A

          \[\leadsto \mathsf{fma}\left(\frac{1}{v}, \left(m \cdot \color{blue}{\left(\left(\mathsf{neg}\left(m\right)\right) + 1\right)}\right) \cdot m, \mathsf{neg}\left(m\right)\right) \]

      19. distribute-lft-in N/A

          \[\leadsto \mathsf{fma}\left(\frac{1}{v}, \color{blue}{\left(m \cdot \left(\mathsf{neg}\left(m\right)\right) + m \cdot 1\right)} \cdot m, \mathsf{neg}\left(m\right)\right) \]

      20. *-rgt-identity N/A

          \[\leadsto \mathsf{fma}\left(\frac{1}{v}, \left(m \cdot \left(\mathsf{neg}\left(m\right)\right) + \color{blue}{m}\right) \cdot m, \mathsf{neg}\left(m\right)\right) \]

      21. lower-fma.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{1}{v}, \color{blue}{\mathsf{fma}\left(m, \mathsf{neg}\left(m\right), m\right)} \cdot m, \mathsf{neg}\left(m\right)\right) \]

      22. lower-neg.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{1}{v}, \mathsf{fma}\left(m, \color{blue}{\mathsf{neg}\left(m\right)}, m\right) \cdot m, \mathsf{neg}\left(m\right)\right) \]

      23. lower-neg.f64 87.4

          \[\leadsto \mathsf{fma}\left(\frac{1}{v}, \mathsf{fma}\left(m, -m, m\right) \cdot m, \color{blue}{-m}\right) \]

   4. Applied rewrites 87.4%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{1}{v}, \mathsf{fma}\left(m, -m, m\right) \cdot m, -m\right)} \]

   5. Taylor expanded in m around 0

      \[\leadsto \color{blue}{m \cdot \left(\frac{m}{v} - 1\right)} \]
   6. Step-by-step derivation

      1. distribute-lft-out-- N/A

         \[\leadsto \color{blue}{m \cdot \frac{m}{v} - m \cdot 1} \]

      2. associate-/l* N/A

         \[\leadsto \color{blue}{\frac{m \cdot m}{v}} - m \cdot 1 \]

      3. unpow2 N/A

         \[\leadsto \frac{\color{blue}{{m}^{2}}}{v} - m \cdot 1 \]

      4. *-rgt-identity N/A

         \[\leadsto \frac{{m}^{2}}{v} - \color{blue}{m} \]

      5. lower--.f64 N/A

         \[\leadsto \color{blue}{\frac{{m}^{2}}{v} - m} \]

      6. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{{m}^{2}}{v}} - m \]

      7. unpow2 N/A

         \[\leadsto \frac{\color{blue}{m \cdot m}}{v} - m \]

      8. lower-*.f64 85.1

         \[\leadsto \frac{\color{blue}{m \cdot m}}{v} - m \]

   7. Applied rewrites 85.1%

      \[\leadsto \color{blue}{\frac{m \cdot m}{v} - m} \]
   8. Step-by-step derivation

   9. Applied rewrites 97.4%

      \[\leadsto \mathsf{fma}\left(\frac{m}{v}, \color{blue}{m}, -m\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 97.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \cdot \left(\frac{m \cdot \left(1 - m\right)}{v} + -1\right) \leq -2 \cdot 10^{+37}:\\ \;\;\;\;\frac{m \cdot \left(m \cdot m\right)}{-v}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{m}{v}, m, -m\right)\\ \end{array} \]
5. Add Preprocessing

Reproduce

herbie shell --seed 2024227 
(FPCore (m v)
  :name "a parameter of renormalized beta distribution"
  :precision binary64
  :pre (and (and (< 0.0 m) (< 0.0 v)) (< v 0.25))
  (* (- (/ (* m (- 1.0 m)) v) 1.0) m))